Maximum Likelihood Estimation of a Likelihood Ratio Ordered Family of Distributions

We consider bivariate observations $(X_1,Y_1), \ldots, (X_n,Y_n)\subset \mathfrak{X}\times\mathbb{R}$ with a real set $\mathfrak{X}$ such that, conditional on the $X_i$, the $Y_i$ are independent random variables with distribution $P_{X_i}$, where $(P_x)_{x\in\mathfrak{X}}$ is unknown. Using an empirical likelihood approach, we devise an algorithm to estimate the unknown family of distributions $(P_x)_{x\in\mathfrak{X}}$ under the sole assumption that this family is increasing with respect to likelihood ratio order. We review the latter concept and realize that no further assumption such as all distributions $P_x$ having densities or having a common countable support is needed. The benefit of the stronger regularization imposed by likelihood ratio ordering over the usual stochastic ordering is evaluated in terms of estimation and predictive performances on simulated as well as real data.